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Several non-standard problems for the stationary Stokes system. (English) Zbl 1437.35586
Summary: This paper studies the Stokes system $$-\Delta{\mathbf{u}}+\nabla\rho={\mathbf{f}}$$, $$\nabla\cdot{\mathbf{u}}=\chi$$ in $$\Omega$$ with three boundary conditions: ${\mathbf{n}}\cdot{\mathbf{u}}={{\mathbf{n}}\cdot\mathbf{g}}, \quad{\mathbf{n}}\cdot{\mathbf{u}}={\mathbf{n}}\cdot\mathbf{g}, \quad\displaystyle{\mathbf{n}}\cdot{\mathbf{u}}={{\mathbf{n}}\cdot\mathbf{g}}.$ Here $$\Omega$$ is a bounded simply connected planar domain. We find a necessary and sufficient condition for the existence of a solution in Sobolev spaces $$W^{s,q}(\Omega;{\mathbb{R}}^2)\times W^{s-1,q}(\Omega)$$, with $$1+1/q<s<\infty$$, in Besov spaces $$B_s^{q,r}(\Omega;{\mathbb{R}}^2)\times B_{s-1}^{q,r}(\Omega)$$, with $$1+1/q<s<\infty$$, and classical solutions in $$\mathcal{C}^{k,\alpha}(\overline{\Omega},{\mathbb{R}}^2)\times{\mathcal{C\%}}^{k-1,\alpha}(\overline{\Omega})$$, with $$0<\alpha<1$$, $$k\in{\mathbb{N}}$$.
##### MSC:
 35Q35 PDEs in connection with fluid mechanics 76D07 Stokes and related (Oseen, etc.) flows 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35A09 Classical solutions to PDEs 76D10 Boundary-layer theory, separation and reattachment, higher-order effects
##### Keywords:
Stokes system; Navier-type boundary conditions
Full Text:
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